976 
It is, however, immediately evident that dy, is no covariant vector 
though g,, is one, so that we should compare with Haminton’s 
invariant integral another, which is no longer invariant. 
The same difficulty arises if a virtual displacement of the. gravi- 
tational field is defined as the shift of a set of values ga, from the 
point-instant 2, to another next to it ze, + C7’. By so doing we do 
not obtain a covariant variation 
Oda b 
Ogar = — ZP) dr | 
vp 
12. A closer examination of the geometrical meaning of the 
tensor components gq, teaches us that in virtue of the equation 
e? = (ab) qq, dea des they form together an infinitesimal quadratic 
three-dimensional extension, the ‘‘indicatrix”’ around each point-instant 
of the field figure. 
The whole gravitational tield may be said to be represented by the 
totality of the indicatrices described around the different point- 
instants, in the same way as in elementary considerations an elec- 
tric field is described by Farapay’s lines of force. A virtual displa- 
cement of the gravitation field must therefore mean a displacement 
of all these indicatrices, in such a way, that it does not disturb the 
configuration and intersections of the indicatrices. 
Let us consider two neighbouring indicatrices h and j, which 
intersect in the figure 7. We may give the displacements to the 
indicatrix 4 and the indicatrix 7 separately and also to the figure 7. 
We then demand that the shifted figure 7 shall again be the inter- 
section of the shifted indicatrices Jh’ and /’. 
This cannot be managed by the variation specified in the preced- 
ing paragraph. There all point-instants of an indicatrix were 
supposed to undergo one and the same virtual displacement, equal 
to that which belongs to the centre. Now on the contrary we require 
that the virtual displacements of the point-instants of an indicatrix 
be defined by the values of dr’ at the different point-instants them- 
selves, . 
If the Sv are not constant, the virtual displacement will generally 
consist not only in a certain translation, but also in a rotation of 
the indicatrices. Analogous considerations may be applied to the 
virtual displacement of the electro-magnetic field. The potentials 
which together form a covariant tensor of the first order, represent 
at each point-instant a trivector multiplied by V—g, 1. e. (in 
infinitesimal dimensions) a certain linear three-dimensional extension. 
